Showing papers on "Dual norm published in 2009"
Posted Content•
TL;DR: This work describes and analyzes a systematic method for constructing matrix-based regularization techniques and demonstrates the potential of this framework by deriving novel generalization and regret bounds for multi-task learning, multi-class learning, and multiple kernel learning.
Abstract: There is growing body of learning problems for which it is natural to organize the parameters into matrix, so as to appropriately regularize the parameters under some matrix norm (in order to impose some more sophisticated prior knowledge). This work describes and analyzes a systematic method for constructing such matrix-based, regularization methods. In particular, we focus on how the underlying statistical properties of a given problem can help us decide which regularization function is appropriate.
Our methodology is based on the known duality fact: that a function is strongly convex with respect to some norm if and only if its conjugate function is strongly smooth with respect to the dual norm. This result has already been found to be a key component in deriving and analyzing several learning algorithms. We demonstrate the potential of this framework by deriving novel generalization and regret bounds for multi-task learning, multi-class learning, and kernel learning.
157 citations
TL;DR: In this paper, the authors studied the correlation of these norms with certain algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit.
Abstract: A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on ℤ/Nℤ introduced by Gowers in his proof of Szemeredi’s Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg’s proof of Szemeredi’s Theorem) defined by the authors. For each integer k ≥ 1, we define seminorms on l∞(ℤ) analogous to these norms and seminorms. We study the correlation of these norms with certain algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences). Using these seminorms, we define a dual norm that acts as an upper bound for the correlation of a bounded sequence with a nilsequence. We also prove an inverse theorem for the seminorms, showing how a bounded sequence correlates with a nilsequence. As applications, we derive several ergodic theoretic results, including a nilsequence version of the Wiener-Wintner ergodic theorem, a nil version of a corollary to the spectral theorem, and a weighted multiple ergodic convergence theorem.
93 citations
TL;DR: The norm of composition followed by differentiation operator from the Bloch and the little Bloch space to the weighted space H μ ∞ on the unit disk is calculated.
88 citations
01 Nov 2009
TL;DR: Norm Derivative Characterizations of Inner Product Spaces Orthogonality Relations Norm Derivatives and Heights Perpendicular Bisectors in Real Normed Spaces Bisectrices in RealNormed Spaces Areas of Triangles in Normed Real Spaces as discussed by the authors
Abstract: Norm Derivatives Characterizations of Inner Product Spaces Orthogonality Relations Norm Derivatives and Heights Perpendicular Bisectors in Real Normed Spaces Bisectrices in Real Normed Spaces Areas of Triangles in Normed Real Spaces.
68 citations
Posted Content•
04 Oct 2009
TL;DR: A single inequality is isolated which seamlessly implies both generalization bounds and on-line regret bounds; and it is shown how to construct strongly convex functions over matrices based on strongly conveX functions over vectors.
Abstract: There is growing body of learning problems for which it is natural to organize the parameters into matrix, so as to appropriately regularize the parameters under some matrix norm (in order to impose some more sophisticated prior knowledge). This work describes and analyzes a systematic method for constructing such matrix-based, regularization methods. In particular, we focus on how the underlying statistical properties of a given problem can help us decide which regularization function is appropriate. Our methodology is based on the known duality fact: that a function is strongly convex with respect to some norm if and only if its conjugate function is strongly smooth with respect to the dual norm. This result has already been found to be a key component in deriving and analyzing several learning algorithms. We demonstrate the potential of this framework by deriving novel generalization and regret bounds for multi-task learning, multi-class learning, and kernel learning.
63 citations
TL;DR: In this paper, the authors give a complete picture of the boundedness and compactness of the products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces of holomorphic functions on the unit disk.
Abstract: We give a complete picture of the boundedness and compactness of the products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces of holomorphic functions on the unit disk.
60 citations
TL;DR: In this article, the general solution and the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary -norms were obtained.
Abstract: We obtain the general solution and the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary -norms .
55 citations
TL;DR: Upper bounds for the dual norms of residuals that are explicit in terms of local Poincare constants are derived that are illustrated for negative first order Sobolev norms and a dual norm arising in convection-reaction-diffusion problems.
Abstract: We derive upper bounds for the dual norms of residuals that are explicit in terms of local Poincare constants. Residuals are continuous linear functionals that are orthogonal to a finite element space and have a singular part supported on the skeleton of the underlying mesh. Functionals of this type play a key role in a posteriori error estimation. Our main tools are a discrete partition of unity and suitably weighted trace and Poincare inequalities. The technique is illustrated for negative first order Sobolev norms and a dual norm arising in convection-reaction-diffusion problems.
42 citations
TL;DR: In this article, duality results for adjoint operators and product norms in the framework of Euclidean spaces were proved and used to derive condition numbers especially when perturbations on data are measured componentwise relatively to the original data.
Abstract: We prove duality results for adjoint operators and product norms in the framework of Euclidean spaces. We show how these results can be used to derive condition numbers especially when perturbations on data are measured componentwise relatively to the original data. We apply this technique to obtain formulas for componentwise and mixed condition numbers for a linear function of a linear least squares solution. These expressions are closed when perturbations of the solution are measured using a componentwise norm or the infinity norm and we get an upper bound for the Euclidean norm.
30 citations
TL;DR: In this paper, it was shown that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional, and that the dual of any lush infinite-dimensional real space contains a copy of the Daugavet property.
Abstract: We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker) alternative Daugavet property. We construct a (noncomplete) strictly convex predual of an infinite-dimensional $L_1$ space (which satisfies a property called lushness which implies numerical index $1$). On the other hand, we show that a lush real Banach space is neither strictly convex nor smooth, unless it is one-dimensional. Therefore, a rich subspace of the real space $C[0,1]$ is neither strictly convex nor smooth. In particular, if a subspace $X$ of the real space $C[0,1]$ is smooth or strictly convex, then $C[0,1]/X$ contains a copy of $C[0,1]$. Finally, we prove that the dual of any lush infinite-dimensional real space contains a copy of $\ell_1$.
26 citations
TL;DR: In this paper, it was shown that a Gruenhage compact space admits an equivalent, strictly convex dual norm if and only if ϒ is a tree and ϒ = span ¯.
Journal Article•
TL;DR: In this article, the authors studied the correlation of norms on Z/NZ with algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences).
Abstract: A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on Z/NZ introduced by Gowers in his proof of Szemeredi's Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg's proof of Szemeredi's Theorem) defined by the authors. For each integer k >= 1, we define seminorms on l(infinity)(Z) analogous to these norms and seminorms. We study the correlation of these norms with certain algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences). Using these seminorms, we define a dual norm that acts as an upper bound for the correlation of a bounded sequence with a nilsequence. We also prove an inverse theorem for the seminorms, showing how a bounded sequence correlates with a nilsequence. As applications, we derive several ergodic theoretic results, including a nilsequence version of the Wiener-Wintner ergodic theorem, a nil version of a corollary to the spectral theorem, and a weighted multiple ergodic convergence theorem.
TL;DR: In this paper, the dual norm of d ( 2 ) ( w, q ) and the James constant of d( 2 ) w, q ) (w, q) were determined.
Abstract: In [M. Kato and L. Maligranda, On James and Jordan–von Neumann constants of Lorentz sequence spaces, J. Math. Anal. Appl. 258 (2001) 457–465], it is an open problem to compute the James constant of the dual space of two dimensional Lorentz sequence space d ( 2 ) ( w , q ) . In this paper, we shall determine the dual norm of d ( 2 ) ( w , q ) and completely compute the James constant of d ( 2 ) ( w , q ) .
TL;DR: In this paper, it was shown that if 1 1 > 0, the operator I + T attains its norm, and if ∥I + T∥ > 1, then the operator does not attain its norm.
Abstract: It is shown that if 1 1, the operator I + T attains its norm. A reflexive Banach space X and a bounded rank one operator T on X are constructed such that ∥I + T∥ > 1 and I + T does not attain its norm.
TL;DR: In this article, it was shown that the positive operators from X to Y are elements of the cone LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone.
Abstract: If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖+‖, q(y) = ‖y
+‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases.
TL;DR: In this article, it was shown that for every e> 0e very Banach space with a w ∗ -separable dual, there exists a 1+e-equivalent norm with the ball covering property.
Abstract: A normed space is said to have ball-covering property if its unit sphere can be contained in the union of countably many open balls off the origin. This paper shows that for every e> 0e very Banach space with a w ∗ -separable dual has a 1+e-equivalent norm with the ball covering property.
TL;DR: In this paper, the authors present equivalent conditions for a space $X$ with an unconditional basis to admit an equivalent norm with a strictly convex dual norm, where the dual norm is a convex regularization.
Abstract: We present equivalent conditions for a space $X$ with an unconditional basis to admit an equivalent norm with a strictly convex dual norm.
01 Jan 2009
TL;DR: In this article, the authors present new convergence results and new versions of Fatou lemma in Mathematical Economics based on various tightness conditions and the existence of scalarly integrable selections theorems for the (sequential)-weak-star upper limit of a sequence of measurable multifunctions taking values in the dual Eof a separable Banach space.
Abstract: We present new convergence results and new versions of Fatou lemma in Mathematical Economics based on various tightness conditions and the existence of scalarly integrable selections theorems for the (sequential)-weak-star upper limit of a sequence of measurable multifunctions taking values in the dual Eof a separable Banach space E. Existence of conditional expectation of weakly-star closed random sets in a non norm separable dual space is also provided.
TL;DR: In this paper, upper and lower bounds for the p-angular distance in normed linear spaces are given. But none of the obtained upper bounds are better than the corresponding results due to L. Maligranda recently established in the paper======[simple norm inequalities, Amer. Math. Monthly, 113(2006), 256-260].
Abstract: New upper and lower bounds for the p-angular distance in normed
linear spaces are given. Some of the obtained upper bounds are better than the
corresponding results due to L. Maligranda recently established in the paper
[Simple norm inequalities, Amer. Math. Monthly, 113(2006), 256-260].
01 Jan 2009
TL;DR: In this paper, rigorous a posterioriL2 error bounds for reduced basis approximations of the unsteady viscous Burgers’ equation in one space dimension are presented.
Abstract: In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers' equation in one space dimension. The a poste- riori error estimator, derived from standard analysis of the error{residual equation, comprises two key ingredients | both of which admit ecient Oine{Online treatment: the rst is a sum over timesteps of the square of the dual norm of the residual; the second is an accurate upper bound (computed by the Successive Constraint Method) for the exponential{in{time stability factor. These error bounds serve both Oine for construction of the reduced basis space by
TL;DR: For the class of transformers acting as X → ∫ Ω A t X B t d μ ( t ) on the space of bounded Hilbert space operators, the authors gave formulae for its norm on the Hilbert-Schmidt class.
TL;DR: It is shown that the functional a posteriori error estimates provide sharp upper and lower bounds of the error and their practical computation requires solving only finite-dimensional problems.
Abstract: We present new a posteriori error estimates for the finite volume approximations of elliptic problems. They are obtained by applying functional a posteriori error estimates to natural extensions of the approximate solution and its flux computed by the finite volume method. The estimates give guaranteed upper bounds for the errors in terms of the primal (energy) norm, dual norm (for fluxes), and also in terms of the combined primal-dual norms. It is shown that the esti- mates provide sharp upper and lower bounds of the error and their practical computation requires solving only finite-dimensional problems.
TL;DR: In this paper, it was shown that a metric space X has weak finite decomposition complexity with respect to the operator norm localization property if and only if X itself has the operator-norm localization property.
TL;DR: The C∗-valued norm is defined on a Hilbert C ∗ -module by its standard inner product as discussed by the authors and generalizations of a number of classical inequalities known for either complex numbers or Hilbert space operators.
Abstract: The C∗ -valued norm is defined on a Hilbert C∗ -module by its standard inner product. In this paper we give generalizations of a number of classical inequalities known for either complex numbers or Hilbert space operators. In particular, we study Bohr’s inequality for the C∗ -valued norm on a Hilbert C∗ -module. Mathematics subject classification (2000): primary: 46L08, secondary: 47A63, 26D07, 26D15.
TL;DR: In this paper, the local and global two-weight norm inequalities for solutions to the nonhomogeneous -harmonic equation for differential forms were proved and the weighted Lipschitz norm and BMO norm inequalities were obtained.
Abstract: We first prove the local and global two-weight norm inequalities for solutions to the nonhomogeneous -harmonic equation for differential forms. Then, we obtain some weighed Lipschitz norm and BMO norm inequalities for differential forms satisfying the different nonhomogeneous -harmonic equations.
TL;DR: The dual X ∗ of a Banach space admits a dual σ-LUR norm if and only if X admits a σ weak Kadets norm and moreover X is σ asplund generated.
Posted Content•
TL;DR: In this article, the conditions under which a closed set in a normed linear space is proximinal or Chebyshev are studied, and a part of approximation theory is studied.
Abstract: In this paper, we study a part of approximation theory that presents the conditions under which a closed set in a normed linear space is proximinal or Chebyshev.
TL;DR: In this article, the authors characterize the self-mappings of a holomorphic function in the open unit disc D which preserve the set of all evaluation functionals in the holomorphic space.